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We develop a high order accurate numerical method for solving the elastic wave equation in second-order form. We hybridize the computationally efficient Cartesian grid formulation of finite differences with geometrically flexible discontinuous Galerkin methods on unstructured grids by a penalty based technique. At the interface between the two methods, we construct projection operators for the pointwise finite difference solutions and discontinuous Galerkin solutions based on piecewise polynomials. In addition, we optimize the projection operators for both accuracy and spectrum. We prove that the overall discretization conserves a discrete energy, and verify optimal convergence in numerical experiments.

The diffusive viscous wave equation describes wave propagation in diffusive and viscous media. Examples include seismic waves traveling through the Earth's crust, taking into account of both the elastic properties of rocks and the dissipative effects due to internal friction and viscosity; acoustic waves propagating through biological tissues, where both elastic and viscous effects play a significant role. We propose a stable and high-order finite difference method for solving the governing equations. By designing the spatial discretization with the summation-by-parts property, we prove stability by deriving a discrete energy estimate. In addition, we derive error estimates for problems with constant coefficients using the normal mode analysis and for problems with variable coefficients using the energy method. Numerical examples are presented to demonstrate the stability and accuracy properties of the developed method.

High order upwind summation-by-parts finite difference operators have recently been developed. When combined with the simultaneous approximation term method to impose boundary conditions, the method converges faster than using traditional summation-by-parts operators. We prove the convergence rate by the normal mode analysis for such methods for a class of hyperbolic partial differential equations. Our analysis shows that the penalty parameter for imposing boundary conditions affects the convergence rate for stable methods. In addition, to solve problems with discontinuous data, we extend the method to also have the weighted essentially nonoscillatory property. The overall method is stable, achieves high order accuracy for smooth problems, and is capable of solving problems with discontinuities.

We develop a hybrid spatial discretization for the wave equation in second order form, based on high-order accurate finite difference methods and discontinuous Galerkin methods. The hybridization combines computational efficiency of finite difference methods on Cartesian grids and geometrical flexibility of discontinuous Galerkin methods on unstructured meshes. The two spatial discretizations are coupled with a penalty technique at the interface such that the overall semidiscretization satisfies a discrete energy estimate to ensure stability. In addition, optimal convergence is obtained in the sense that when combining a fourth order finite difference method with a discontinuous Galerkin method using third order local polynomials, the overall convergence rate is fourth order. Furthermore, we use a novel approach to derive an error estimate for the semidiscretization by combining the energy method and the normal mode analysis for a corresponding one-dimensional model problem. The stability and accuracy analysis are verified in numerical experiments.

In this paper, we develop a provably energy stable and conservative discontinuous spectral element method for the shifted wave equation in second order form. The proposed method combines the advantages and central ideas of the following very successful numerical techniques: the summation-by-parts finite difference method, the spectral method, and the discontinuous Galerkin method. We prove energy stability and the discrete conservation principle and derive error estimates in the energy norm for the (1+1)-dimensions shifted wave equation in second order form. The energy-stability results, discrete conservation principle, and the error estimates generalize to multiple dimensions using tensor products of quadrilateral and hexahedral elements. Numerical experiments, in (1+1)-dimensions and (2+1)-dimensions, verify the theoretical results and demonstrate optimal convergence of L2 numerical errors at subsonic, sonic and supersonic regimes.

We develop a stable finite difference method for the elastic wave equation in bounded media, where the material properties can be discontinuous at curved interfaces. The governing equation is discretized in second order form by a fourth or sixth order accurate summation-by-parts operator. The mesh size is determined by the velocity structure of the material, resulting in nonconforming grid interfaces with hanging nodes. We use order-preserving interpolation and the ghost point technique to couple adjacent mesh blocks in an energy-conserving manner, which is supported by a fully discrete stability analysis. In our previous work for the wave equation, two pairs of order-preserving interpolation operators were needed when imposing the interface conditions weakly by a penalty technique. Here, we only use one pair in the ghost point method. In numerical experiments, we demonstrate that the convergence rate is optimal and is the same as when a globally uniform mesh is used in a single domain. In addition, with a predictor-corrector time integration method, we obtain time stepping stability with stepsize almost the same as that given by the usual Courant-Friedrichs-Lewy condition.

We develop a new finite difference method for the wave equation in second order form. The finite difference operators satisfy a summation-by-parts (SBP) property. With boundary conditions and material interface conditions imposed weakly by the simultaneous-approximation-term (SAT) method, we derive energy estimates for the semi-discretization. In addition, error estimates are derived by the normal mode analysis. The proposed method is termed as energy-based because of its similarity with the energy-based discontinuous Galerkin method. When imposing the Dirichlet boundary condition and material interface conditions, the traditional SBP-SAT discretization uses a penalty term with a mesh-dependent parameter, which is not needed in our method. Furthermore, numerical dissipation can be added to the discretization through the boundary and interface conditions. We present numerical experiments that verify convergence and robustness of the proposed method.

In this article, a novel implementation of a widely used pseudo-two-dimensional (P2D) model for lithium-ion battery simulation is presented with a transmission line circuit structure. This implementation represents an interplay between physical and equivalent circuit models. The discharge processes of an LiNi0.33Mn0.33Co0.33O2-graphite lithium-ion battery under different currents are simulated, and it is seen the results from the circuit model agree well with the results obtained from a physical simulation carried out in COMSOL Multiphysics, including both terminal voltage and concentration distributions. Finally we demonstrated how the circuit model can contribute to the understanding of the cell electrochemistry, exemplified by an analysis of the overpotential contributions by various processes.

We consider finite difference approximations of the second derivative, exemplified in Poisson's equation, the heat equation, and the wave equation. The finite difference operators satisfy a summation-by-parts (SBP) property, which mimics the integration-by-parts principle. Since the operators approximate the second derivative, they are singular by construction. When imposing boundary conditions weakly, these operators are modified using simultaneous approximation terms. The modification makes the discretization matrix nonsingular for most choices of boundary conditions. Recently, inverses of such matrices were derived. However, for problems with only Neumann boundary conditions, the modified matrices are still singular. For such matrices, we have derived an explicit expression for the Moore-Penrose inverse, which can be used for solving elliptic problems and some time-dependent problems. For this explicit expression to be valid, it is required that the modified matrix does not have more than one zero eigenvalue. This condition holds for the SBP operators with second and fourth order accurate interior stencil. For the sixth order accurate case, we have reconstructed the operator with a free parameter and show that there can be more than one zero eigenvalue. We have performed a detailed analysis on the free parameter to improve the properties of the second derivative SBP operator. We complement the derivations by numerical experiments to demonstrate the improvements.